What is the current procedure under Conservative Party rules for kicking out a leader who doesn't want to go?

British political parties, at least the major ones, are historically not very tolerant of a leader who blows an election as May did this one.

The leader in these cases often resigns or possibly faces a back bench revolt.

Yeah, but what is the actual mechanism? In Australia, not too long ago, the caucus of either major party in Parliament could move to hold a leadership election at any time, and just vote in a new leader in one day, so a coup could happen pretty much instantly. This is no longer the case for the Labor Party, so a coup would now be a more drawn out, bloody affair, and it's harder to get rid of a struggling leader.

What about the UK? IIRC, in Thatcher's day it was pretty easy, but now it's become more complicated, because the wider membership of the party is involved as well?

The problem is that a leadership challenge could take months while the Brexit negotiations already started. May probably will resign if there is a leadership challenge with significant backbench support, but that would lead to other people jumping in. And I don't see a 2016 scenario where May was the overwhelming favorite while Leadsom stumbled from gaffe to gaffe and had almost no support in the parliamentary party. An election with Davis, Johnson, Hammond and Rudd probably won't end up like a walk-over.

As far as I know, the primary procedure should May refuse to go would be a leadership challenge for party leader.

That's not a real answer.

I'm asking what is the leadership challenge procedure. How does it work? In some parliamentary systems, you just hold a vote of the MPs of that party, and it's over in a day. In others, it's a more drawn out affair, that involves the wider party membership, and can take months. I think for the UK Conservative Party, it's the latter, but I'm asking if someone can offer specifics.

Also, is there any precedent for a party winning a plurality of seats in a UK election, and forming the government (even in coalition with another party), yet still dumping the PM immediately after the election?